Optimization of an integrated feedback control for a pest management predator-prey model

Zhenzhen Shi; Huidong Cheng; Yu Liu; Yanhui Wang; Zhenzhen Shi; Huidong Cheng; Yu Liu; Yanhui Wang

doi:10.3934/mbe.2019401

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7963-7981. doi: 10.3934/mbe.2019401

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Optimization of an integrated feedback control for a pest management predator-prey model

1.
College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
2.
College of Foreign Languages, Shandong University of Science and Technology, Qingdao 266590, Shandong, China

Received: 04 March 2019 Accepted: 28 August 2019 Published: 02 September 2019

In this paper, a Leslie-Gower predator-prey model with ratio-dependence and state pulse feedback control is established to investigate the effect of spraying chemical pesticides and supplement amount of beneficial insects at the same time. Firstly, the existence, uniqueness and asymptotic stability of the periodic solution are proved by using successor functions method and the analogue of the Poincaré criterion when the equilibria points $E_{*}$ and $E_{0}$ are stable, and the existence of the limit cycle without impulse system are verified when the equilibrium $E_{*}$ is the unstable point. Furthermore, to obtain the minimum cost per period of controlling pests, we propose the optimization problem and calculate the optimal threshold. Finally, the feasibility of our model is proved by numerical simulation of a concrete example.
- semi-continuous dynamic systems,
- order-1 periodic solution,
- successor functions,
- limit cycle,
- optimization
Citation: Zhenzhen Shi, Huidong Cheng, Yu Liu, Yanhui Wang. Optimization of an integrated feedback control for a pest management predator-prey model[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7963-7981. doi: 10.3934/mbe.2019401

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Abstract

In this paper, a Leslie-Gower predator-prey model with ratio-dependence and state pulse feedback control is established to investigate the effect of spraying chemical pesticides and supplement amount of beneficial insects at the same time. Firstly, the existence, uniqueness and asymptotic stability of the periodic solution are proved by using successor functions method and the analogue of the Poincaré criterion when the equilibria points $E_{*}$ and $E_{0}$ are stable, and the existence of the limit cycle without impulse system are verified when the equilibrium $E_{*}$ is the unstable point. Furthermore, to obtain the minimum cost per period of controlling pests, we propose the optimization problem and calculate the optimal threshold. Finally, the feasibility of our model is proved by numerical simulation of a concrete example.

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